Math 2.0

Field theory and polynomials Corollary

Degree of a simple extension is the degree of the minimal polynomial

Builds on K(a) is isomorphic to K[x] modulo the minimal polynomialDegree of a field extensionMinimal polynomial of an algebraic element

Statement

Let L/KL/K be a field extension and let a∈La \in L be algebraic over KK with minimal polynomial mam_a of degree nn. Then [K(a):K]=n[K(a) : K] = n, and 1,a,a2,…,an−11, a, a^2, \dots, a^{n-1} is a basis of K(a)K(a) over KK.

Remarks

The proof: by the structure theorem, K(a)≅K[x]/(ma)K(a) \cong K[x]/(m_a). By polynomial division, the classes of 1,x,…,xn−11, x, \dots, x^{n-1} are a basis of K[x]/(ma)K[x]/(m_a). For example, [ℚ(23):ℚ]=3[\mathbb{Q}(\sqrt[3]{2}) : \mathbb{Q}] = 3, with the basis 1,23,431, \sqrt[3]{2}, \sqrt[3]{4}. With the tower law, the degree of every element of a finite extension L/KL/K divides [L:K][L : K].

Reading path

To understand this page from the ground up, read these 51 pages in order. The foundations come first.

  1. Group concept
  2. Ring definition
  3. Commutative ring definition
  4. Ideal definition
  5. Polynomial ring definition
  6. Ring homomorphism definition
  7. Subring definition
  8. Unit of a ring definition
  9. Zero divisor definition
  10. Divisibility in a commutative ring definition
  11. Ideal generated by a subset definition
  12. Maximal ideal definition
  13. Two-sided ideal definition
  14. Degree of a polynomial definition
  15. Polynomial evaluation definition
  16. Field definition
  17. Integral domain definition
  18. Kernel of a ring homomorphism definition
  19. Ring isomorphism definition
  20. Principal ideal definition
  21. Quotient ring definition
  22. The only ideals of a field are zero and the whole field statement
  23. Evaluation homomorphism definition
  24. Leading coefficient definition
  25. Root of a polynomial definition
  26. Subfield definition
  27. Irreducible element definition
  28. Euclidean domain definition
  29. Correspondence theorem for rings statement
  30. Principal ideal domain definition
  31. Monic polynomial definition
  32. Universal property of the quotient ring statement
  33. Irreducible polynomial definition
  34. Degree of a product statement
  35. Field extension definition
  36. R/M is a field iff M is maximal statement
  37. In a PID a nonzero element is irreducible iff it generates a maximal ideal statement
  38. Division algorithm for polynomials statement
  39. Algebraic element definition
  40. R[x] is an integral domain iff R is statement
  41. First isomorphism theorem for rings statement
  42. Degree of a field extension definition
  43. Intermediate field definition
  44. Minimal polynomial of an algebraic element definition
  45. K[x] is a Euclidean domain statement
  46. Field generated by adjoining elements definition
  47. The minimal polynomial divides every polynomial vanishing at the element statement
  48. K[x]/(f) is a field iff f is irreducible statement
  49. Simple extension definition
  50. K(a) is isomorphic to K[x] modulo the minimal polynomial statement
  51. Degree of a simple extension is the degree of the minimal polynomial statement

Where this leads

6 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Simple_extension, Definition
About this page
Tag0N5QW9K7MH
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Kindstatement, corollary
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UsesK(a) is isomorphic to K[x] modulo the minimal polynomial, Degree of a field extension, Minimal polynomial of an algebraic element

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